Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. A B C D Chap 7-1 Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: X μ P(x) i N .3 18 20 22 24 21 4 σ (X μ) i N .2 .1 0 2 2.236 18 20 22 24 A B C D x Uniform Distribution Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-2 Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 _ P(X) .3 .2 .1 0 18 19 20 21 22 23 24 _ X (no longer uniform) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-3 Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: μX X i 18 19 19 24 21 σX N 16 2 ( X μ ) i X N (18 - 21)2 (19 - 21)2 (24 - 21)2 1.58 16 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-4 Comparing the Population Distribution to the Sample Means Distribution Population N=4 μ 21 σ 2.236 Sample Means Distribution n=2 μX 21 σ X 1.58 _ P(X) .3 P(X) .3 .2 .2 .1 .1 0 18 20 22 24 A B C D Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. X 0 18 19 20 21 22 23 24 _ X Chap 7-5 Sample Mean Sampling Distribution: Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) σ σX n Note that the standard error of the mean decreases as the sample size increases Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-6 Sample Mean Sampling Distribution: If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with μX μ Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. and σ σX n Chap 7-7 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of X : Z where: ( X μX ) σX ( X μ) σ n X = sample mean μ = population mean σ = population standard deviation n = sample size Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-8 Sampling Distribution Properties μx μ (i.e. x is unbiased ) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Normal Population Distribution μ x μx x Normal Sampling Distribution (has the same mean) Chap 7-9 Sampling Distribution Properties (continued) As n increases, Larger sample size σ x decreases Smaller sample size μ Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. x Chap 7-10 Sample Mean Sampling Distribution: If the Population is not Normal We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution: μx μ Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. and σ σx n Chap 7-11 Central Limit Theorem As the sample size gets large enough… n↑ the sampling distribution becomes almost normal regardless of shape of population x Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-12 Sample Mean Sampling Distribution: If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency μx μ σ σx n Variation Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. μ x Sampling Distribution (becomes normal as n increases) Larger sample size Smaller sample size μx x Chap 7-13 How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 For normal population distributions, the sampling distribution of the mean is always normally distributed Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-14 Example Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2? Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-15 Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of approximately normal x is … with mean μx = 8 σ 3 …and standard deviation σ x n 36 0.5 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-16 Example (continued) Solution (continued): 7.8 - 8 X -μ 8.2 - 8 P(7.8 X 8.2) P 3 σ 3 36 n 36 P(-0.4 Z 0.4) 0.3108 Population Distribution ??? ? ?? ? ? ? ? ? μ8 Sampling Distribution Standard Normal Distribution Sample .1554 +.1554 Standardize ? X Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. 7.8 μX 8 8.2 x -0.4 μz 0 0.4 Z Chap 7-17